Theorem 1.1 ([1]})   
For any positive integer \(k\) , every \(n\) -node graph has a \((2k-1)\) -spanner on \(O(n^{1+1/k})\)  edges.
Moreover, assuming the girth conjecture [2]}, this is best possible in the sense that there are \(n\) -node graphs that do not admit \((2k-1)\) -emulators on \(o(n^{1+1/k})\)  edges.

